The Indian context is somewhat unusual for understanding the U.S. labor market, but arguably that throws the idea into all-the-stronger relief. Here is an abstract and paper from Supreet Kaur, who is on the job market from Harvard this year:
Wage and employment responses to rainfall shocks in 500 Indian districts from 1956-2008 provide evidence for downward nominal wage rigidity in markets for casual daily agricultural labor. First, nominal wages rise in response to positive labor demand shocks but do not fall during droughts. Second, after transitory positive shocks have dissipated, nominal wages do not return to their previous levels—they remain high in future years. Third, inflation moderates these effects: when inflation is higher, real wages are more likely to be lower during droughts and after transitory positive shocks. Fourth, wage distortions generate employment distortions: employment is lower in the year after a transitory positive shock than if the positive shock had not occurred. Those with less land—who must sell their labor to other farms—are considerably more likely to face rationing. Landless laborers experience a 7% reduction in employment—twice as large as the employment decrease during a drought. Fifth, there is some evidence that wages are less rigid in areas where rigidity is likely to cause larger profit losses due to crop characteristics. Finally, data from a new survey I conducted in two Indian states suggests that agricultural workers and employers: view nominal wage cuts as unfair; are considerably less likely to regard real wage cuts as unfair if they are achieved through inflation rather than nominal cuts; and believe that nominal wage cuts cause effort reductions.
It is impressive how Kaur focuses on the cross-sectional variation. You will note, of course, that wage rigidity for established workers is not the same as wage rigidity for the currently unemployed, or wage rigidity for new job market entrants, who also have high rates of unemployment.